Maximum likelihood estimation for nearly non-stationary stable autoregressive processes

The maximum likelihood estimate (MLE) of the autoregressive coefficient of a near-unit root autoregressive process Y _t=ρ _nY _t-1+e{open} _t with α-stable noise {e{open} _t} is studied in this paper. Herein ρ _n=1-γ/n, γ≥0 is a constant, Y ₀ is a fixed random variable and ε _t is an α-stable random variable with characteristic function φ(t,θ) for some parameter θ. It is shown that when 0<α<1 or α>1 and Ee{open} ₁=0, the limit distribution of the MLE of ρ _n and θ are mixtures of a stable process and Gaussian processes. On the other hand, when α>1 and Ee{open} ₁≠0, the limit distribution of the MLE of ρ _n and θ are normal. A Monte Carlo simulation reveals that the MLE performs better than the usual least squares procedures, particularly for the case when the tail index α is less than 1. © 2011 Blackwell Publishing Ltd.